Let ABC be an acute- angled triangle and AD, BE, and CF be its medians, where E and F are at (3,4) and (1,2) respectively. The centroid of `DeltaABC` ,`G(3,2)`.
The coordinates of D are

Options

A. (7,-4)
B. (5,0)
C. (7,4)
D. (-3,0)

Explanation

To find the coordinates of point D in triangle ABC, we will use the properties of the centroid and the coordinates of the other points provided.

### Step-by-Step Solution:

1. **Understanding the Centroid Formula**:
The centroid \( G \) of a triangle with vertices at \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by the formula:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Here, we know the coordinates of the centroid \( G(3, 2) \).

2. **Assigning Coordinates**:
- Let the coordinates of point D be \( (x, y) \).
- The coordinates of points E and F are given as:
- \( E(3, 4) \)
- \( F(1, 2) \)

3. **Setting Up the Equations**:
Using the centroid formula, we can set up two equations based on the x-coordinates and y-coordinates:
- For the x-coordinates:
\[ 3 = \frac{x + 3 + 1}{3} \]
- For the y-coordinates:
\[ 2 = \frac{y + 4 + 2}{3} \]

4. **Solving the x-coordinate Equation**:
Multiply both sides of the x-coordinate equation by 3:
\[ 9 = x + 3 + 1 \]
Simplifying gives:
\[ 9 = x + 4 \]
Therefore:
\[ x = 9 - 4 = 5 \]

5. **Solving the y-coordinate Equation**:
Multiply both sides of the y-coordinate equation by 3:
\[ 6 = y + 4 + 2 \]
Simplifying gives:
\[ 6 = y + 6 \]
Therefore:
\[ y = 6 - 6 = 0 \]

6. **Final Coordinates of D**:
Thus, the coordinates of point D are:
\[ D(5, 0) \]

### Summary:
The coordinates of point D are \( (5, 0) \).